On the CNOT-Cost of TOFFOLI Gates

The three-input TOFFOLI gate is the workhorse of circuit synthesis for classical logic operations on quantum data, e.g., reversible arithmetic circuits. In physical implementations, however, TOFFOLI gates are decomposed into six CNOT gates and several one-qubit gates. Though this decomposition has been known for at least 10 years, the authors provide here the first demonstration of its CNOT-optimality. They study three-qubit circuits which contain less than six CNOT gates and implement a block-diagonal operator, then show that they implicitly describe the cosine-sine decomposition of a related operator.